Foundations of Stochastic Inventory Theory

by Evan Porteus

Overview

In 1958, Stanford UniversityPress published Studies in the Mathematical Theory of Inventory and Production (edited by Kenneth J. Arrow, Samuel Karlin, and Herbert Scarf), which became the pioneering road map for the next forty years of research in this area. One of the outgrowths of this research was development of the field of supply-chain management, which deals with the ways organizations can achieve competitive advantage by coordinating the activities involved in creating products—including designing, procuring, transforming, moving, storing, selling, providing after-sales service, and recycling. Following in this tradition, Foundations of Stochastic Inventory Theory has a dual purpose, serving as an advanced textbook designed to prepare doctoral students to do research on the mathematical foundations of inventory theory and as a reference work for those already engaged in such research.

The author begins by presenting two basic inventory models: the economic order quantity model, which deals with "cycle stocks," and the newsvendor model, which deals with "safety stocks." He then describes foundational concepts, methods, and tools that prepare the reader to analyze inventory problems in which uncertainty plays a key role. Dynamic optimization is an important part of this preparation, which emphasizes insights gained from studying the role of uncertainty, rather than focusing on the derivation of numerical solutions and algorithms (with the exception of two chapters on computational issues in infinite-horizon models).

All fourteen chapters in the book, and four of the five appendixes, conclude with exercises that either solidify or extend the concepts introduced. Some of these exercises have served as Ph.D. qualifying examination questions in the Operations, Information, and Technology area of the Stanford Graduate School of Business.

Product Details

ISBN-13:	9780804743990
Publisher:	Stanford University Press
Publication date:	08/09/2002
Edition description:	1
Pages:	320
Product dimensions:	6.38(w) x 9.25(h) x 0.90(d)

About the Author

Evan L. Porteus is the Sanwa Bank Professor of Management Science at the Stanford Graduate School of Business.

Table of Contents

	Preface	xv
	Conventions	xviii
1	Two Basic Models	1
1.1	The EOQ Model	1
1.2	The Newsvendor Model	7
	Exercises	16
	References	25
2	Recursion	27
2.1	Solving a Triangular System of Equations	27
2.2	Probabilistic Analysis of Models	28
2.3	Proof by Mathematical Induction	29
2.4	Shortest-Route Problems	29
2.5	Stochastic Shortest-Route Problems	32
2.6	Deterministic Production Planning	34
2.7	Knapsack Problems	35
	Exercises	36
	References	40
3	Finite-Horizon Markov Decision Processes	41
3.1	Example: e-Rite-Way	42
3.2	General Vocabulary and Basic Results	47
	Exercises	54
	References	56
4	Characterizing the Optimal Policy	57
4.1	Example: The Parking Problem	57
4.2	Dynamic Inventory Management	64
4.3	Preservation and Attainment	72
	Exercises	73
	References	76
5	Finite-Horizon Theory	77
5.1	Finite-State and -Action Theory	77
5.2	Proofs for the Finite-State and -Action Case	83
5.3	Generalizations	86
5.4	Optimality of Structured Policies	87
	Exercises	88
	References	90
6	Myopic Policies	91
6.1	General Approaches to Finding Solutions	92
6.2	Development	93
6.3	Application to Inventory Theory	96
6.4	Application to Reservoir Management	97
6.5	Extensions	98
	Exercises	100
	References	102
7	Dynamic Inventory Models	103
7.1	Optimality of (s, S) Inventory Policies	103
7.2	Linear-Quadratic Model	111
	Exercises	115
	References	118
8	Monotone Optimal Policies	119
8.1	Intuition	119
8.2	Lattices and Submodular Functions	122
8.3	A Dynamic Case	126
8.4	Capacitated Inventory Management	128
	Exercises	131
	References	132
9	Structured Probability Distributions	133
9.1	Some Interesting Distributions	133
9.2	Quasi-K-Convexity	137
9.3	A Variation of the (s, S) Inventory Model	139
9.4	Generalized (s, S) Policies	143
	Exercises	148
	References	150
10	Empirical Bayesian Inventory Models	151
10.1	Model Formulation	152
10.2	Conjugate Priors	155
10.3	Scalable Problems	159
10.4	Dimensionality Reduction	161
	Exercises	163
	References	166
11	Infinite-Horizon Theory	167
11.1	Problem Formulation	167
11.2	Mathematical Preparations	170
11.3	Finite State and Action Theory	173
11.4	Generalizations	178
	Exercises	178
	References	180
12	Bounds and Successive Approximations	181
12.1	Preliminary Results	182
12.2	Elimination of Nonoptimal Actions	184
12.3	Additional Topics	188
	Exercises	190
	References	192
13	Computational Markov Decision Processes	193
13.1	Policy Iteration	193
13.2	Use of Linear Programming	195
13.3	Preparations for Further Analysis	197
13.4	Convergence Rates for Value Iteration	199
13.5	Bounds on the Subradius	201
13.6	Transformations	202
	Exercises	207
	References	208
14	A Continuous Time Model	209
14.1	A Two-Product Production/Inventory Model	210
14.2	Formulation and Initial Analysis	211
14.3	Results	216
	Exercises	221
	References	221
Appendix A	Convexity	223
A.1	Basic Definitions and Results	223
A.2	Role of the Hessian	230
A.3	Generalizations of Convexity	234
	Exercises	235
	References	239
Appendix B	Duality	241
B.1	Basic Concepts	241
B.2	The Everett Result	245
B.3	Duality	250
	Exercises	255
	References	260
Appendix C	Discounted Average Value	261
C.1	Net Present Value	262
C.2	Discounted Average Value	264
C.3	Alternatives with Different Time Horizons	267
C.4	Approximating the DAV	268
C.5	Application to the EOQ Model	271
C.6	Random Cycle Lengths	273
C.7	Random-Yield EOQ Problem	274
	Exercises	275
	References	278
Appendix D	Preference Theory and Stochastic Dominance	279
D.1	Basic Concepts	280
D.2	Stochastic Dominance	282
D.3	How to Define Variability	286
D.4	Application to the Newsvendor Problem	287
	Exercises	289
	References	291
	Index	293

Editorial Reviews

"Evan Porteus has organized the important foundations that are essential not only to appreciate this area and apply known results effectively, but that are crucial to fuel further progress that inevitably will be required to meet new practical challenges."

Carnegie Mellon University - Sridhar Tayur

"This book provides a comprehensive foundation for inventory theory. Amazingly, Evan Porteus is able to introduce and explain very complex concepts in simple and interesting ways. In providing the intuitions behind many of the theories and results, he brings difficult concepts to life. This is a must-read for anyone who wants to learn, apply, or conduct research in stochastic inventory theories."

Stanford University - Hau Lee

"Every organization is striving to match supply with demand. Foundations of Stochastic Inventory Theory introduces the fundamental theories for tackling this challenging management task. Emphasizing simple, intuitive, and practical inventory policies rather than complex theories for general settings, Evan Porteus has written both a great textbook for graduate students in management, as well as a great reference book for anyone interested in inventory theory. Simply put, this is one of the best books on inventory theory available."

Yale School of Management - Lode Li

"This book provides a powerful and insightful approach to the analysis and control of stochastic dynamic systems. The introduction to dynamic optimization is focused and efficient with emphasis on how the theory can be applied to operational control settings such as inventory management and many others. At last, with this book every student and researcher interested in optimal operational control can benefit!"

Northwestern University - Jan A. Van Mieghem

"This book provides a comprehensive foundation for inventory theory. Amazingly, Evan Porteus is able to introduce and explain very complex concepts in simple and interesting ways. In providing the intuitions behind many of the theories and results, he brings difficult concepts to life. This is a must-read for anyone who wants to learn, apply, or conduct research in stochastic inventory theories."—Hau Lee, Stanford University


"Every organization is striving to match supply with demand. Foundations of Stochastic Inventory Theory introduces the fundamental theories for tackling this challenging management task. Emphasizing simple, intuitive, and practical inventory policies rather than complex theories for general settings, Evan Porteus has written both a great textbook for graduate students in management, as well as a great reference book for anyone interested in inventory theory. Simply put, this is one of the best books on inventory theory available."—Lode Li, Yale School of Management


"Evan Porteus has organized the important foundations that are essential not only to appreciate this area and apply known results effectively, but that are crucial to fuel further progress that inevitably will be required to meet new practical challenges."—Sridhar Tayur, Graduate School of Industrial Administration, Carnegie Mellon University


"This book provides a powerful and insightful approach to the analysis and control of stochastic dynamic systems. The introduction to dynamic optimization is focused and efficient with emphasis on how the theory can be applied to operational control settings such as inventory management and many others. At last, with this book every student and researcher interested in optimal operational control can benefit!"—Jan A. Van Mieghem, Kellogg School of Management, Northwestern University

From the Publisher

Emphasizing the foundations of dynamic optimization models, this book presents foundational results, concepts, methods, and tools useful in building and analyzing stochastic inventory models. Topics are included according to their application to particular inventory models. There is no attempt to present a comprehensive or totalizing view of stochastic inventory theories. Porteus teaches management science at the Stanford Graduate School of Business. Annotation c. Book News, Inc., Portland, OR